Correlation functions in the Non Perturbative Renormalization Group and field expansion

The usual procedure of including a finite number of vertices in Non Perturbative Renormalization Group equations in order to obtain n-point correlation functions at finite momenta is analyzed. This is done by exploiting a general method recently introduced which includes simultaneously all vertices although approximating their momentum dependence. The study is performed using the self-energy of the tridimensional scalar model at criticality. At least in this example, low order truncations miss quantities as the critical exponent η by as much as 60%. However, if one goes to high order truncations the procedure seems to converge rapidly.     

Integration par parties dans l'espace de Wiener et approximation du temps local

Summary Let be a real-valued stochastic process having a continuous local timeL(u,t),u —, 0tT andX (t) = ( *X)(t),t 0, the regularization ofX by means of the convolution with the approximation of unity . The main theorem in this paper (Theorem 3.5) is a generalization of various results about the approximation (for fixedu) of the local timeL(u, ) by means of a convenient normalization of the numberN ^X (u;) of crossings of the processX with the levelu. Especially, this Theorem extends to a class of not necessarily Markovian continuous martingales, a result of this type for one-dimensional diffusions due to Azais [A2]). The methods of proof combine some estimations of the moments of the number of crossings with a level of a regular stochastic processes with stochastic analysis techniques based upon integration by parts in the Wiener space.

---

---

Support partill du CICYT, N^o PB86-0238     

A numerical algorithm for zero counting. II: Distance to ill-posedness and smoothed analysis

We show a Condition Number Theorem for the condition number of zero counting for real polynomial systems. That is, we show that this condition number equals the inverse of the normalized distance to the set of ill-posed systems (i.e., those having multiple real zeros). As a consequence, a smoothed analysis of this condition number follows.     

On the Expected Condition Number of Linear Programming Problems

Summary.   Let A be an n×m real matrix and consider the linear conic system

Sur le recouvrement du cercle par des ensembles placés au hasard

ItT is the unit circle and (B _n ) a given sequence of subsets ofT, we consider the problem of coveringT with random independent uniform translates of (B _n ). The technique used is based on a theorem on rearrangements of functions.